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A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path.
Hamiltonian if it has a Hamiltonian cycle. Closure: The (Hamiltonian) closure of a graph G, denoted Cl(G), is the simple graph obtained from G by repeatedly adding edges joining pairs of nonadjacent vertices with degree sum at least jV(G)j until no such pair remains. Lemma 10.1 A graph G is Hamiltonian if and only if its closure is Hamiltonian.
A Hamiltonian path is a traversal of a (finite) graph that touches each vertex exactly once. If the start and end of the path are neighbors (i.e. share a common edge), the path can be extended to a cycle called a Hamiltonian cycle .
Definition: Closure. Let \(G\) be a graph on \(n\) vertices. The closure of \(G\) is the graph obtained by repeatedly joining pairs of nonadjacent vertices \(u\) and \(v\) for which \(d(u) + d(v) ≥ n\), until no such pair exists.
Hamiltonian Cycles and Paths. Let G be a graph. A cycle in G is a closed trail that only repeats the rst and last vertices. A Hamiltonian cycle (resp., a Hamiltonian path) in G is a cycle (resp., a path) that visits all the vertices of G.
The hamiltonian problem; determining when a graph contains a spanning cycle, has long been fundamental in Graph Theory. Named for Sir William Rowan Hamilton, this problem traces its origins to the 1850’s.
A closure of a graph G, C(G), is the graph with vertex set V (G) obtained from G by iteratively adding edges joining pairs of nonadjacent vertices whose degrees sum to at least n, until no such pair remains. A graph G is Hamiltonian if and only if its closure C(G) is Hamiltonian. The closure of G is also well-de ned.
